Founded a century ago, Sigma Delta Tau is a historically Jewish sorority with an interest in philanthropy. Their slogan is “empowering women,” and in some way that I cannot possibly understand, this is achieved partly through extremely detailed guidelines on attire. However, the slide above does make a good case study in computational anthropology.

Whenever we have a set of OK/Not OK pronouncements like these, decision tree learning is a good tool for extracting the underlying pattern (I used the rpart package in R). For colours, we can perform analysis using hue, saturation, and value. In this case, the first restriction computed by the tool (and reinforced in the text of the slide) is “don’t go too light” – the sorority requires a colour value below about 79. The second restriction in the decision tree is “not too blue” – specifically a hue lower than about 182. Saturation is not identified as important in the decision tree analysis.

The diagram below highlights the acceptable colour region and the specific examples from the slide above. Of course, this only gives clarity to what the social rule is. It does not explain why the social rule exists, or what social goals the rule might achieve. For that, we must turn to traditional anthropology – although even here, social simulation can provide computational assistance.

There are three main (though closely related) branches of mathematics – the study of number, the study of shape, and the study of relationships. An interesting ethnomathematical example of the latter is the “skin group” system of the Lardil people of Mornington Island in Australia.

Lardil children

Similar systems (often with two groups, called moieties) can be found around the world (I first discovered the concept as a child, in the classic young adult science fiction novel Citizen of the Galaxy). Among the Lardil, there are eight groups, each associated with particular totemic creatures or objects:

Skin Group

Name

Totem

1

Burulungi

Lightning

2

Ngariboolungi

Shooting star

3

Bungaringi

Turtle

4

Yugumari

Seagull

5

Gungulla

Grey shark

6

Bulunyi

Crane

7

Bulyarini

Sea turtle

8

Gumerungi

Rock

Membership of a “skin group” implies a complex set of tribal obligations and taboos, but the most significant is that only certain kinds of marriages are permitted. Members of group 1 must marry people from group 2 (and vice versa), and similarly for the pairs 3/4, 5/6, and 7/8. All other marriages are considered to be incestuous.

We can define a mathematical function, the spouse function σ, that maps each person’s “skin group” to the “skin group” that their spouse must have: σ(1) = 2, σ(2) = 1, σ(3) = 4, σ(4) = 3, etc. For each of the eight kinds of valid marriage, there is also a rule for determining the “skin group” of the children:

Father

Mother

Children

1

2

8

2

1

3

3

4

2

4

3

5

5

6

4

6

5

7

7

8

6

8

7

1

We can define two mathematical functions, the father-of function φ and the mother-of function μ, that map the “skin group” of a father or mother to the “skin group” that their children must have: φ(1) = 8, μ(1) = 3, φ(2) = 3, μ(2) = 8, etc.

This is all much clearer when displayed visually. In the diagram, two-part black arrows →→ indicate valid marriages. The arrows run from the “skin group” of the wife to the “skin group” of the husband. Red arrows → run from each marriage arrow to the “skin group” of the children. Together, the arrows form an octagonal prism:

Following a single black arrow and then a red arrow (→→) gives the mother-of function μ, with μ(1) = 3, etc. It can be seen that this function has a four-generation cycle: μ(μ(μ(μ(x)))) = x or, as it is often expressed, μ4(x) = x. In other words, each person’s “skin group” is the same as that of their great-great-grandmother in the female line (the maternal grandmother of their maternal grandmother).

Following a single black arrow backwards (from the head end) and then a red arrow (←→) gives the father-of function φ, with φ(1) = 8, etc. It can be seen that this function has a two-generation cycle: φ(φ(x)) = x or, as it is often expressed, φ2(x) = x. In other words, each person’s “skin group” is the same as that of their grandfather in the male line (their paternal grandfather).

The combination of the two cycles makes the Lardil “skin group” system a very effective way of shuffling genes within a small population, thus avoiding inbreeding. It also highlights the fact that mathematics can be found in some surprising places. And there are even more patterns to be found in this example. Among others, σ(x) = φ(μ(x)). Also, μ(φ(μ(x))) = φ(x), which some readers may recognise as indicating a dihedral group.